https://www.khanacademy.org/math/in-sixth-grade-math https://www.khanacademy.org/profile/peterwcollingridge/
http://spacemath.gsfc.nasa.gov/SMBooks/SMEarthV2.pdf
http://spacemath.gsfc.nasa.gov/Modules/6Module1.html http://galileoandeinstein.physics.virginia.edu/lectures/gkastr1.html
http://betterexplained.com/articles/learning-how-to-count-avoiding-the-fencepost-problem/
Chapter 1: Knowing our numbers
operators
Smaller Than / Bigger Than: Arrows!
- The larger number shoots the smaller number
Comparing numbers
- Writing Numbers in Expanded form
- write \(14,987\) in expanded
- \(1 * 10,000 = 10,000\)
- \(4 * 1,000 = 4,000\)
- \(9 * 100 = 900\)
- \(8 * 10 = 80\)
- \(7 * 1 = 7\)
- \(10000+4000+900+80+7 = 14,987\)
Large Numbers in Practice
- In this topic we’ll look at the basics of number approximation, when we round numbers to be less exact
- This would happen when you don’t trust your measurements or want to simplify things, in other words when you want to roughly know the number
Rounding to nearest 10
- What is the nearest multiple of 10 for a given number?
- Look at value in ones column, one place to the right of the tens, and decide which multiple of ten is closest, then Round up or Round down
- multiples of ten = \(10,20,30,40,50,60,70,80,90\)
value:nearest multiple of ten = 36:40, 34:30, 65:70, 92:90, 12:10 
- note: Rule of Rounding 5: When rounding number 5, which is 5 away from one multiple and 5 away from the other, the rule is to round up
Rounding to nearest 100
- Look at value one place to the right of the hundreds column, in the tens column and decide which multiple of hundred is closest.
value:nearest multiple of ten/hundred = \(154:150/200,\\ 4,674:4,670/4,700,\\ 9,995:10,000/10,000,\\ 8,346:8,350/8300\) 
2 step estimation problem
- The Pokemon day-care centre has 128 orphaned Pokemon. They’ve just been given another 54. The daycare wants to share these Pokemon amongst the 43 people who have signed up to take a Pokemon. Roughly speaking how many Pokemon will each trainer get?
- Round 128 and 54 to the nearest ten
- Round the amount of trainers to the nearest ten
- Divide the amount of Pokemon by amount of trainers
- Let
P be amount of Pokemon
- 170 / 40 = P
- P x 40 = 170. Solve for P
- List Multiples of 40: 40, 80, 120, 160, 200.
- 170/40 = 4 Pokemon. Each Trainer will receive roughly 4 Pokemon.
Multiplication estimation example
- A ticket agent sells 42 tickets. Tickets cost £28 each. Use rounding to estimate the total amount of money from sales
- Round number of tickets and cost of tickets to nearest ten
- Multiply together
- 4030 = 43 with two zeroes added on the end (12 -> 1,200)
- Roughly we will have £1,200 from sales
Using Brackets
Constructing Numerical Expressions with Parathesis
- Mike and Amon are having a beyblade battle. Amon has 3 Beyblades and wins 1 more. He then battles Mike for all or nothing, doubling his amount of Beyblades. Write an expression to model this situation without performing any operations
Chapter 2: Whole Numbers
Properties of Whole Numbers
Commutative Law of addition
- commutative law = order doesn’t matter if you’re adding a bunch of things. i.e. you can add numbers in any order and the result will always be the same
- \(10+20+4=34\)
- \(4+10+20=34\)
Commutative law of multiplcations
- commutative law = numbers can be multiplied in any order and the result will always be the same
Associative law of addition
- associative law of addition = It doesn’t matter which numbers are added first in paranthesis, the result will always be the same
- i.e. it doesn’t matter if you associate the 20 with the 1, or the 1 with the 4.
- (20+1)+4 = 25
- (1+4)+20 = 25
Associative law of multiplication
- Associative law of multiplication = It doesn’t matter which numbers are added first in brackets, the result will always be the same
Distributive property over addition
- distributive law of multiplication over addition = distributive the number you are multiplying by to each number in the brackets
- 4(8+3). Distribute the 4 to each number
- (48)+(43)
- 32+12 = 44
- Why use distributive law instead of just BODMAS (adding numbers in brackets first)?
- Using the distributive law helps you in algebra, when you will have variables
- example 1:
- 4(8+x)=44. Solve for
x
- (48) + (4x) = 44
- 32 + 4x = 44
- 4x = 12
- x = 3
- example 2:
- Each month you pay the following bills: Water:$30, Electricity:$150, Cable:$70. How much do you pay in a year and how much is each item?
- 12(30w,150e,70c)
- 360w + 1800e + 840c = $3000.
- We could of added the bills together, then multiplied [12(250) = 3000], but you would of lost crucial information about how much each bill costs
- Related Branches: Factoring, Polynomials, Algebra. Knowing the distributive law will be useful when you reach those branches
Distrbutive property over substraction
- distributive law of multiplication over substraction = distributive the number you are multiplying by to each number in the brackets
- \(5(9-4)\)
- \(5*9-5*4\)
- \(45-20\)
- \(25\)
Summary
- Commumative Law = a+b+c = b+c+a
- Associative law = a+(b+c) = b+(a+c)
- Distributive property = a+b(c+d) = a+bc+bd
Chapter 3: Playing with Numbers
- In this chapter we will learn about factors and multiples and about the divisibility rules for 2, 3, 4, 5, 6, 8, 9 and 10
Factors and Multiples
- In this topic we will see that a factor of a number is an exact divisor of that number, and a multiple is a result of the multiplication of that number
Finding Factors of a Number
- In this tutorial we’ll begin to look at the numbers that “make up” the number
Use: This will be useful throughout maths, whether we are adding up fractions, exploring mystical numbers patterns or breaking computer codes, factoring numbers are key!
- factor = the factory that makes a whole number from parts
- note: factor comes from latin and makes do/make. The factor makes something happen. In math, a factor makes a result happen, it produces the result
- i.e. 2 and 6 make 12 happen / 2 and 6 produce a result of 12 (2*6= 12)
- Finding Factors
- Write the smallest and largest factor
- Work up the number line(1:10), filling in the gaps between the smallest and largest factor
- Keep going until the gaps are filled
- Recognizing Divisibility: is x divisible by…
- Number 1. All Whole Numbers are divisble by 1
- Number 2. The number is even (i.e, the last digit is 2, 4, 6, 8, 0)
- Number 3. The digits add up to a multiple of 3.
- If you’re unsure if the sum is multiple of 3, you can check the sum it self by adding the digits together.
- I.e. Is 386, 802 divisble by 3? 3+8+6+8+0+2=27. Not sure if 27 is multiple of 3? 2+7=9. Yes, 9 (therefore 27) is multiple of 3.
- Number 4. The last two digits are a multiple of 4
- Number 5. The last digit is zero or 5
- Number 6. Divisible by both 2 and 3
- Number 7.
- Number 8. The last three digits are divisble by 8
- Number 9. The digits add up to a multiple of 9
- Number 10. The last digit is zero
- Find All Factors of 120
- \(120 = 1 * 120\)
- \(120 = 2 * 60\). Is x divisible by 2? Check if one’s place is even.
- \(120 = 3 * 40\). Is x divisible by 3? Add up its digits and see if sum is divisible by three. I.e. 1+2+0 = 3. Yes, 120 is divisible by 3.
- \(120 = 4 * 30\) Is x divisible by 4? You ignore everything beyond hundreth’s place and look at last two digits.i.e., is 20 divisible by 4, so 120 is also divisible by 4.
- \(120 = 5 * 24\) Is x divisible by 5? If last digit is 0 or 5, then yes
- \(120 = 6 * 20\). Is x divisible by 6? If divisible by 2 and 3, yes
- \(120 = 8 * 15\) Is x divisible by 8? If last 3 digits divisble by 8, yes
- \(120 = 10 * 12\) Is x divisble by 10? If last digit is 0, then yes
Factors of 120 are: 1 2 3 4 5 6 8 10 12 15 20 24 30 40 60 120
FUN <- function(x) {
# Convert number to integer to speed up function
x <- as.integer(x)
# Find all numbers to divide by (1:x)
div <- seq_len(abs(x))
# Only return numbers where division results in no remainder
# 0L, the `L` notation ensures number is stored a a integer
# not a double
factors <- div[x %% div == 0L]
# Make a list with negative and positive factors
factors <- list(neg = -factors, pos = factors)
return(factors)
}
Find Multiples Of A Number
- To find a multiple the best way is just simply list the multiples by doing your times table
- factor x factor = multiple
Prime and Composite Numbers
- In this topic we will see that composite numbers have more than one factor and prime numbers have just two: 1 and the number itself
- Prime numbers have been studied by mathematicians and mystics for eons. By studying them you will unfold more fascinating views of the universe
- use: Prime numbers are used in cryptography and studying nature, it’s what decides how many petals a flower has and what stops theives from entering your online bank account
Prime Numbers
- Prime numbers are the building blocks of numbers, that can not be broken down into products of smaller numbers
- prime number = A natural number (1,2,3…) divisible by exactly two natural numbers, 1 and itself
- 1 = not prime. Only divisible by one number, not exactly two
- 2 = prime number. Is divisble by 1 and it self
- 2 is the only even number that is prime
- Identifying Prime Numbers
- Divide number by all the numbers between 1 and it self. If the number is only divisible by 1 and it self then it is prime
FUN <- function(x) {
# Convert number to integer to speed up function
x <- as.integer(x)
# Find all numbers to divide by (1:x)
div <- seq_len(abs(x))
# Only return numbers where division results in no remainder
# 0L, the `L` notation ensures number is stored a a integer
# not a double
factors <- div[x %% div == 0L]
# Check If amount of factors is larger than 2
if (length(factors) > 2) {
print(paste(x,"is not prime"))
} else {
return(factors)
}
}
# Which of these numbers is prime?
sapply(c(35,47,55,60,87), FUN)
Test for Divisbillity of numbers
The why of the three divisibillity
- Take Number, such as 498
- Re-write number as something that is “1 + something divisble by 3”"
- 498 -> 4(1+99) + 9(1+9) + 8
- Instead of writing 4(100)…the trick is write 4(1+something-divisible-by-3) instead of 100
- Distribute the four and re-arrange terms
- 4+(499)+9+(99)+8.
- 4(99) + 9(9) + 4 + 9 + 8. # Re-arrange the terms
- Check if terms are divisible by 3
- 4(99), divisible by 3 because anything multipled by something divisible by three is divisible by three!
- 9(99), divisible by 3, because anything multiplied by something divisible ny three is divisble by three!
- 4(99) + 9(99), If you addtwo things which are divisible by three added, the whole thing is still going to by divisble by three
- 4 + 9 + 8 = 21. 2+1. What are we left with? Our original digits! Just need to make sure original digits are divisble by three
- In other words, 498 is divisble by 3 if 4 + 9 + 8 is divisible by 3!
The Why of the 9 divisbillity rule?
- Break the number up by place value
- 2(1000) + 9(100) + 4(10) + 3
- Re-write as “1 + something divisble by 9”
- 2(1+999)+9(1+99)+4(1+9)+3
- Distribute
- 2+(2999)+9+(999)+4+(9*4)+3
- Re-arrange the terms (2999)+(999)+(94)+2+9+4+3
- Check divisibillity of all parts
- (2999)+(999)+(9*4) = yes. Anything times something divisible by 9 is divisble by 9
- 2+9+4+3 = yes. Second part needs to be divisble by 9 in order for the whole number (2943) to be divisible by 9.
- In other words, 2943 is divisble by 9 if 2+9+4+3 is divisible by 9! ### The why of the 4 divisibillity rule
- is 419108 disivible by 4?
- A number is disible by 4 if the last two digits are divisble by 4
We-can re-write the number as a multiple of 100 plus the last two digits
- 419108 = 4191 + 08 Because 419100 is a multiple of 100, it is also a multiple of 4
So long as the last two-digits are divisble by four then the original number must be divisble b four!
Common Divisibillity
- Concepts = prime factors, division
- All numbers divisible by A and B are also divisible by…
- example All numbers divisble by 21 and 20 are also divisble by…
- Find Prime Factors for 21 and 20
- 21 = 7 * 3. 20 = 522
- Any number divisble by both 21 and 20 must have 22357 as part of its prime factorization
- Check if the prime factors of the options match
- The prime factors of 30 is 235
- Since 235 is a part of 22357, all numbers divisble by 21 and 20 must be divisible by 30.
Prime Factorization
You know what prime numbers are and how to identify them, in this tutorial we will look at how all positive numbers can be broken down into products of prime numbers (In some ways, prime numbers as the atoms of the number world that can be multiplied to create any number!). Besides being a fascinating idea, it’s extremely useful! Prime factorization can be used to decrypt encrypted information.
Concepts = division, prime factors.
Finding the prime factors
Find the prime factorisation of 75. Write your answer using exponential notation.
- Work through lists of primes, finding the smallest prime number that will go into 75.
- note: this is where learning about divisibillity rules comes in handy
- list of primes = 2,3,5,7,11…
- Is 75 divisible by 2? No. Ones place is not even.
- is 75 divisible by 3? Yes. 5+7=12, which is divisble by 3.
- 75 = 3 * 25
- Find the factors of 25.
- is 25 divisble by 2? No. If it’s not divisble by 75, then won’t be by 25.
- is 25 divisble by 3? No. 2+5 = 3.
- is 25 divisble by 5? Yes.
- 25 = 5 * 5
- We’re done, we’re left with only primes!
- 75 = 3 * 5 * 5
- Re-write using exponential notation
- If we have repeated primes we can write those as exponents
- 5*5 is the same thing as 5^2
- 75 = 3 * 5^2
Lowest Common Multiple
Life is good but it can be better. Just imagine being able to find the smallest number that is a multiple of two other numbers! Other than making your life more fufilling, it will allow you to do incredible things like adding fractions
concepts = Prime Factors, Fractions, Addition
use =
- Adding fractions
- Wastage. For example, if buns come in packs of ten and burgers in packs of 8. If you don’t want to waste any you will need to buy the LCM of buns and burgers. e.g. LCM(10,8). This also occurs in manufacturing, where one company will only give you parts in packages of X and another will only give you parts in packes of Y. If you need one of each to make your product you will need to find the LCM(X,Y) to avoid waste parts.
- Finding best deals.
- Method 1: Prime Factorization
- What is the LCM of A and B?
- Take Prime Factorization of A and B and construct smallest number whose ingredients is all the factors of both A and B, which will be the LCM
- What is the LCM of 12 and 18?
- Find prime Factors of 12 and 18 using factorization trees
- Multiply of the factors that appear in both numbers and construct a new number
- The LCM has a super-set containing all of these factors or all of these factors in it as many times as we have it in any one of these (A or B)
- in order to be divisible by 12 you need to have two 2’s and atleast one 3. But to be disivisible by 18 you need another 3! 233*2 is divisible by both 12 and 18.
- LCM(18,12) = 233*2 = 36
- Method 2: Brute Force
- Go through times table for both numbers until you find LCM
Method 1 (Prime Factorization) is better as firstly, you deconstruct the number (so it’s fun!) and secondly, things will get hairer with larger numbers so it’s easier, quicker and cleaner to have a systematic method

- note: if two numbers do not share any common factors, i.e. they are both primes, then their LCM is a product of the two
- What is the LCM of 11 and 13?
- 11 and 13 do not share any common factors, so LCM is product of both
- LCM = 11 * 13 = 143 ## Greatest Common Factor
You know how to find factors of a number. What about factors that are common to two numbers? Even better, imagine the largest factors that are common to two numbers.
note: It’s better to just use method 2.
- What is the GCF of A and B?
- Finding the GCF/GCD(greatest common divisor) is finding the largest number that divides into both numbers
- method 1: Common Factors:
- List all factors for each number, and find the greatest common factor
- What is the GCF of 10 and 7?
- 10 = 1,2,5,10
- 7 = 1,7
- GFC(10,7) = 1
- note: two numbers with only 1 as their GCF is known as relatively prime
- Method 2: Common Prime Factors:
- List all prime factors for each number, and find the largest set of common prime factors
- The greatest common factor will be a product of the set
- What is the GCF of 30 and 15?
- 30 = 2,3,5
- 21 = 7,3
- GFC(30,31) = 3
- What is the GCF of 105 and 30?
- Factor each number completely as a product of its primes
- Find the common prime factors
- Largest Set of Common Prime Factors = 3,5
- GCF(105,30) = 3*5 = 15
- 15 is the largest number divisible into both numbers
- note: If numbers do not share any prime factors, then the GCF is 1 as this factor is common to all numbers
- What is the GCF of 60 and 60?
- Factor each number completely as a product of its primes
- 160 = 2,2,3,5
- 60 = 2,2,3,5
- Both numbers share all of their factors, so GCF = 223*5 = 60
- note: In general, the greatest common factor of any number and its self is that number
GCF and LCM word problems
Uses
Factors
GCF
- Creating Identical Packages with two items(A and B)
- Find GCF, divide each item by GCF
- GCF = the greatest number of packages that can be made
- item / GCF = amount of items in each package
- A / GCF = number of item A in each package
- B / GCF = number of item B in each package
- Splitting items A and B into identical rows
- Find GCF, divide each item by GCF
- GCF = the greatest number of rows that can be planted
- item / GCF = amount of items in each row
- A / GCF = number of item A in each package
- B / GCF = number of item B in each package
- Simplify fractions
- Find GCF of num/dom, divide each by GCF
LCM
- LCM is great for finding where two events will occur together, by finding where the multiples of each number first meet
- Preventing Wastage. item A and B is needed to build a product/make a hotdog. You can only buy item A in packs of X and item B in packs of Y.
- Find LCM, divide LCM by each pack
- LCM(X,Y) = smallest total number of item A/B you can buy without wasting.
- I.e. the amount of items you need of A and B each to not have any left-overs.
- LCM / packs = amount of packs you need to buy of each item
- LCM / X = amount of packs you need to buy of item A
- LCM / Y = amount of packs you need to buy of item B
- Finding over-lapping events
- Finding minimum over-lapping scores.
- Pikachu gains 8 points with every match. Squirtle gains 9 points. At the end of the day, each pokemon has the same amount of points.
- Find LCM.
- LCM = Least number of total points that person A and B could of each achieved
- Pikachu gains 8 points with every match. Squirtle gains 9 points. At the end of the day, each pokemon has the same amount of points. What is the least amount of matches Pikachu could of had?
- Find LCM, divide LCM by points per event for each person
- LCM = Least number of total points that person A and B could of each achieved
- LCM / points per event = amount of events that each person did in order to reach the same score
- compare unlike fractions
- Find the lowest common denominator of two unlike fractions, so you can compare them
Chapter 4: Basic Geometrical Ideas
Introduction
- Euclid Father Of Geometery = Euclid, born 2,500 years ago in Alexandira was the first person to prove something beyond doubt. Not simply have a good feeling but prove that something is wholly correct for the whole universe.
While he wasn’t the first to study geometery, many peoples would of before him simply by looking at the shapes around us in nature and considering what shapes will help us build structures, like pots or even Pyramids. However in his book he essentially under-pinned modern maths which is begining with an basic assumption, known as a theorem, and then going to prove it, known as a proof. Thus, Euclid’s legacy lies in turning theorems, stuff you feel pretty sure about, into proofs, stuff you know beyond a shadow of a doubt!
Terms & Labels in Geometery
Measuring Segments
- congruent = two line sigments with the same length
Parallel and perpendicular lines
- parallel = lines that intersect at a right angle (90 degrees). (Makes a X shape)
- perpendicular = two lines that are on the same plane and neve intersect, like rail-roads.
- intersecting lines = lines that intersect but not at 90 degree angles
Identifying parallel and perpendicular lines
- identifying parallel lines = if two suspected parallel lines intersect a third line at the same angle, then they are parallel
Angles
Angles are formed where corners are made.
- angle = when 2 rays share a common endpoint an angle is made. The vertex is the angles heart.
- labelling angles = \(\angle ABC\): A&C = endpoints, B = vertex/the angles’ heart.
- naming angles
- acute = less than 90 degrees
- obtuse = more than 90 degrees
- right angle = 2 rays perpendicular to each other ### Measuring Angles
- angle measurement = how open or closed an angle is
- how to measure: Angles can be measured in degrees or radians
- Degrees comes from the convention of using circles, therefore one method of measuring angles is circle arcs, where a segment is taken of a circle.
- the length of a circle is 360 degrees
- Why is it 360 degrees? No one knows. However, there are hints in history and the way the universe works (sun).
- There are 365 days in a year. 360 is quite close and is divisible by more things, perhaps this is way many ancient astronomers and civilisations (persians/mayans) used 360 day calenders
- The circle being 360 degrees is a convention history has handed us.
- measuring angles in degrees with circle arcs
- Draw a circle with a point in the middle
- Draw a angle (2 rays with a common end point/vertex)
- Make sure the angle’s vertex is placed at the circle’s midpoint
- In the diagram below, the white line is the first ray and the green line is the second ray
- The fraction of the circle’s circumference that is intersected by the 2 rays is the angle’s measurement.
- The segment of the circle that the two rays/angle slices out is the angle’s measurement

Circles
Circles Glossary

Chapter 5: Unerstanding elementary shapes
- corners, edges, planes, open curves and closed curves are different shapes we seen around us. Understanding these shapes can help us solve a wide variety of real life problems
Angles: acute, obtuse and reflex
- acute = angle less than 90 degrees. Rays are close to each other
- right-angle = 90 degrees. Rays are perpendicular to each other
- obtuse = more than 90 degrees.
Measuring Angles
- To review, a circle is split into 360 equal parts. Each part is called a degree, which is one way we measure angles.
Decomposing Angles
- To find out the measure of an angle, you can add or substract other angles

Classifying Triangles

Quadrilaterals
Quadrilateral Glossary
- quadrilateral = any shape with four shapes
- concave = has interior angle larger than 180 degrees. This means none of the sides can be parallel to each other
- note: if angles were equal to 180 degrees, rather than more (concave) or less (convex) then it would be straight lines creating a triangle
- convex = all of the interior angle less than 180 degrees.
- trapezoid = exactly one pair of parallel sides
- parallelogram = two pairs of parallel sides
- rectangle = all 4 angles are a right angles
- rhombus = all 4 sides are equal
- square = rhombus + rectangle
- note: sum of any quadrilateral is 360 degrees

Chapter 6: Integers
Integers
- If you put the whole numbers and the negative numbers together you get a new collection of numbers known as integers.
Negative Integers
- negative number = a number that is less than zero
There are two ways to conceptulize negative numbers. One is a lack of something, such as a lack of temperature (freezing) or a lack of money (debt), the other is distance in relation to zero.
- distance from zero = there are two parts to a number. It’s absolute value and it’s symbol.
- The absolute value tells how far the number is from zero
- The symbol tells you in which direction.
negative = left of zero, positive = right of zero.
lack of something = negative numbers help us model real life situations, such as having debt. If you owe someone $2 and have a balance of $1, then you have a balance of -$1. While -$1 doesn’t actually exist, it does help us keep track of of things, such as owing money or loosing health in a game.
note: there is a difference between the operation of subtraction and the object(a negative number), even though both use the same sign. You should therefore say subtract negative 3 and not minus minus 3 to keep things clear.
- history:
- rules for dealing with negative numbers first appeared in 7th century India, written down by Brahmagupta.
- However, references to negative numbers existed during 200 BCE in China, where positive numbers were items sold (money in) and negative were items brought (money out)
- Greeks did not address negative numbers, as they dealt with geometery. Lengths, areas and volumes of shapes all had to be positive. Futhermore, their ideas were based on magnitudes. Magnitudes were represented by a line or an area, not by a number.
- Negative numbers appeared in Europe by the 15th century, after islamic and byzantine sources were translated
A debt minus zero is a debt.
a negative substract zero is a negative
A fortune minus zero is a fortune.
a positive substract zero is a positive
Zero minus zero is a zero.
zero subtract zero is zero
A debt subtracted from zero is a fortune.
a zero subtract a negative is a positive
A fortune subtracted from zero is a debt. 0-(pos) = neg
a zero subtract a positive is a negative. 0-(neg) = pos
The product of zero multiplied by a debt or fortune is zero.
zero multiplied by a negative or positive is zero
The product of zero multiplied by zero is zero.
zero times zero is zero
The product or quotient of two fortunes is one fortune.
positive * positive = single positive
The product or quotient of two debts is one fortune.
negative * negative = single positive
The product or quotient of a debt and a fortune is a debt.
negative * positive = negative
The product or quotient of a fortune and a debt is a debt.
Addition of Integers with different signs
There are two parts to a addition/subtraction sum. The symbol which represents the object(a positive or negative number) and the symbol which represents the operation(subtraction or addition)
- -2+5 = 3
- Method 1: Go 2 to the left of zero, then 5 to the right
- Method 2: What is the difference between 2 and 5?
Subtraction of integers
Subtracting a negative number = adding a positive number
Negative number word problem: Temperature
- The coldest temperature recorded in england was -27 °C. The warmest temperature 37.4C. What is the difference in temperature?
- To find the difference between two numbers you substract
- 37-(-27) = 64C
- Difference in temperature is 64 degrees
Chapter 7: Fractions
- Fractions represent part of something, part of a whole.
- Natural numbers, 1 2 3 4 5, are good for counting things. But what if something is less than 1? Then we use fractions!
- Fractions are good for measuring and comparing things, while integers are good for counting things.
- Fractions are especially handy in comparing two different total amounts, i.e. if you want to compare the success of schools with a different amount of pupils.
- Fractions break something up into equal parts, like an apple into half (1/2th)
Introduction
- numerator = number of parts/slices/pieces
- on numberline: the amount of steps taken
- denominator = total number of equal parts/pieces/slices the whole is broken/sliced up into
- on numberline: the total amount of steps between 0 and 1
Whole Numbers as Fractions
- A pizza is split into 3 parts. You eat all 3 parts. How much have you eaten?
- Fractions on a number line
- 3/5 = less than one on a number line
- 1/1 = equal to one on the number line
- 3/1 = more than one on a number line, it is 3!
- Notice the numerator is now larger than the denominator
- 1/1 + 1/1 + 1/1 = 3/1
- 4/4 = 1 whole, 1 on a number line
- 4/1 = 4 wholes, 4 on a numbe rline
Recognizing Fractions
- What Fraction does the shaded area represent?
- See how many pieces are shaded in
- Add those pieces together = numerator 3 denominator = the amount of pieces the whole is divded into, even if there is more than one whole shaded in
Fractions on the number line
- Fractions are located between 0 and 1 on the number line
- Unless they’re bigger than 1
- Denomiator: Fractions split the space between 0 and 1 into equal pieces
- 1/2 = splits the number 1 into 2 equal pieces
- 1/5 = splits the number 1 into 5 equal pieces
- The size of each step along the number OR the number of steps between 0 and 1 = denomiator
- Numerator: The posistion on the number line of the fraction is indicated by the numertor
- 1/2 = the first split of the 2 splits
- 1/5 = the first split of the 5 splits
- 3/5 = the third split of the 5 splits
- The number of steps from zero = numerator
- What if the numerator is bigger than the denominator?
- The posistion on the number line is past one
- The posistion carries on taking steps past 1. Each step is still the same length.
- In reality this means you have more than one or the whole. i.e. you have more than one apple pie, you have one apple pie and a extra slice!
- 1/4 = one split into 4 pieces, each located 0.25 places along.
- 2/4 = 0.50. two slies of apple pie
- 3/4 = 0.75. 3 slices of apple pie
- 4/4 = 1. Whole Apple pie
- 5/4 = 1.25. Whole apple pie and one slice.
- 6/4 = 1.50. Whole apple pie and 2 slices.
- examples of fractions on a number line
- 3/8 = The video game is split into 8 levels. There are 8 levels between 0 and 1. 1 = the whole game. You are 3 levels past zero.
- 5/4 = The apple pie is split into four slices. There are 4 pieces between 0 and 1. 1 = a whole pie. You have a whole apple pie and one extra slice.
Improper and Mixed Fractions

- Why does this work?
- example 1: 5 1/4
- The 5 is the same thing as 20/4
- This make the fraction \(20/4 + 1/4 = 21/4\)
- The 5 can be thought as 5 wholes, each split into four pieces. Count the pieces and you get 20 pieces.
- example 2: 3 1/2
- The 3 is the same thing as 6/2
- This make the fraction \(6/2 + 1/2 = 7/2\)
- The 3 can be fought of as 3 wholes, each split into 2 pieces. Count the pieces and you get 6 pieces.

Equivalent Fractions

- Equivalent Fraction Models
- Complete the equation 8/12 = ?/3
- Model A(1/12) has four pieces, for each piece of model B(1/3)
- divide Model A’s numerator and denomator by four = 8/12 -> 2/3
- 8/12 = 2/3
- Complete the equation. 9/12 = ?/8
- Model A(1/12) has 1.5 pieces for each piece of model B(1/8)
- divide numerator and denom of model A by 1.5 = 6/8
- 9/12 = 6/8

Identifying equivalent fractions
- Method One: Reduce both fractions into their simplest forms and see if they are the same
- step one: Reduce both fractions into their simplest form
- Fraction A: Find number that divides both numerator and denominator
- Fraction B: Find number that divides both numerator and denominator
- Step two: Check if the reduced fractions are the same
- If fraction A == fraction B, then fractions are equivalent
- Method two: Intuitive workflow
- step one: Are the denominators the same?
- Yes: Compare Numerators
- No: Can you get the numerators/denominators the same?
- i.e. 30/45 and 54/81. Divide the num/dom of 30/45 by 5 to get 6/9, and divide the nom/dom of 81 by 9 to get 6/9.
- note: if you don’t multiply/divide the numerator/denominator by the same number the value of the fraction will change
- yes: Do the fractions match?
- no: probably not equivalent
Create Equivalent Fractions
- Method 1: Visually imagine both fractions as two equal cakes.
- Cake A is sliced into this many slices and cake B is sliced into this many slices. If this many slices of cake A is eaten, how many slices of cake B needs to be eaten to equal the same amount of total cake?
- \(2/10 = p/100\) What number could replace P to create an equivalent fraction?
- Fraction A represents dividing a rectangular cake into 10 slices and taking 2
- Imagine we cut the cake into 100 slices instead, how many slices would equal the same amount of cake?
- In order the take the same amount of cake, we need to take 20 out of the 100 slices
- Method 2: Creating Equivalent Fractions by multiplying the numerator and denominator by th11e same number
- Another way is to multiply by \(10/10\)
- \(10/10 = 1/1\) so really we are multiplying by one
- \(2/10 = 20/100\)
- Method 3: 2/10 of 100 = 2/10 * 100/1 = 200/10 = 20

Decomposing a mixed number
- Numers ways of visually viewing a mixed number and decomposing it. Note, both fractions/models equal the same value

Like Fractions
- Like fractions are fractions with the same denominator. We will see how to add like fraction together
Adding up Like Fractions
- To add like fractions simply add the numerators together and keep the denominators the same
- example: 1/20 + 20/20 = 21/20. Which is saying a whole pie plus a slice equals a whole pie and a slice
Comparing Fractions
- In like fractions, the fraction with the greatest numerator is largest
- In fractions with the same numerator, the fraction with the greatest denominator is smaller
Comparing fractions with like numerators and denominators
- In like fractions, the fraction with the greatest numerator is largest
- example: Which is largest? 4/7 or 5/7
- re-write: \(4/7 = 4 * 1/7\) and \(5/7 = 5 * 1/7\)
- 4/7 has four 1/7th’s, whereas has 5/7 has five 1/7’ths. 5/7 is larger.
- \(4/7 < 5/7\)
- In fractions with the same numerator, the fraction with the greatest denominator is smaller
- example: Compare 3/4 and 3/9
- re-write: \(3/4 = 3 * 1/4\) and \(3/9 = 3 * 1/9\)
- Which pieces are smaller? \(1/4\) is a square split into four equal pieces, \(1/9\) is a square split into nine equal pieces.
- \(1/4\) is split into fewer pieces, so a fourth is larger than a ninth OR \(1/9\) has smaller pieces, so a ninth is smaller than a fourth
- \(3/4 > 3/9\)
Comparing fractions with unlike numerators and denominators
- To compare unlike fractions, change both fractions to share a common a denomiator (i.e. split all the pies into the same equal pieces)
- step 1: Simplify Fraction (this makes finding the LCM and subsequent multiplying easier )
- step 2: Find Lowest Common Multiple of both denominators, this will be the common denominator
- Prime Factorize Each Denominator, then find the LCM using a superset that contains all prime factors
- step 3: Make both fractions share a common denomiator by multiplying the denominator/numerator of each fraction (multiplying the numerator and denominator of a fraction by the same number does not change the value)
- step 4: Now both fractions have the same denominator, compare the numerator!
- example: compare 1/12 and 4/8
- step 1: Simplify Fraction (this makes finding the LCM and subsequent multiplying easier )
- step 2: Find Lowest Common Multiple of both denominators, this will be the common denominator
- Prime Factorize each and find the superset of primes
- 222*3 = 24.
- 24 is the lowest common multiple / shared denominator
- step 3: Make both fractions share a common denomiator by multiplying the denominator/numerator of each fraction (multiplying the numerator and denominator of a fraction by the same number does not change the value)
- \(1/12 * 2/2 = 2/24\)
- \(4/8 * 3/3 = 12/24\)
- step 4: Now both fractions have the same denominator, compare the numerator!
- \(2/24 < 12/24\)
- \(1/12 < 4/8\)
note: You can often tell intuitively which fraction is smaller/larger by remembering that a larger denominator means that the square is sliced into many and thus smaller equal pieces, and a smaller numerator is opposite, meaning the square is split into fewer and thus larger equal pieces
note: When changing an unlike fraction to a like fraction you are not changing the value of the fractions, but simply how they are represented so it’s easier to compare.
Ordering Fractions
- The easiest way to order a bunch of fractions is to make all fractions like fractions (have the same denominator) and then compare the numerators (to compare exactly, all pies need to be sliced into the same amount of slices)
- To turn unlike fractions into unlike fractions, you need to find a common denominator, by first prime factorizing each fraction’s denominator, then finding the LCM of those denominators (as shown above(“comparing unlike…”) and below)
- example: Order \(3/4, 3/6, 5/12\)
- Turn the unlike fractions into like fractions
- step 1: Prime factorize the denominator of each fraction
- step 2: Find the LCM, using the superset containing all prime factors
- step 3: Change the fractions to have a common denominator by multiplying the num/denom by the same number
- \(3/4 * 3/3 = 9/12\)
- \(3/6 * 2/2 = 6/12\)
- \(5/12 = 5/12\)
- \(3/4 > 3/6 > 5/12\)
- note: an intuitive way to look at it is that 3/4 is more than half the pie, 3/6 is half the pie and 5/12 is less than half the pie
- note: When changing an unlike fraction to a like fraction you are not changing the value of the fractions, but simply how they are represented so it’s easier to compare.
Addition and Subtraction of fractions
Addition of like fractions
- Is fraction mixed? No
- Are the denominators the same? Yes
- Add up numerators, keep denominators the same and simplify!
- \(3/15 + 7/15 = 10/15 = 2/3\)
- simplify: if you changed 5 of the 15th pieces into one piece, how many pieces would you have? 3.
- i.e. if you merged every 5 of the 15 slices into one big slice, how many slices would the pie have? 3 big slices.
Addition of unlike fractions
- Change unlike fractions to like by finding a common denominator via finding the LCM
- LCM = a number that both denominators divide into
- Look at examples in “Comparing unlike fractions/Ordering fractions”
- note: When changing an unlike fraction to a like fraction you are not changing the value of the fractions, but simply how they are represented so it’s easier to compare.
- i.e. if you have 2 identical pies, pie A is sliced into three slices and pie B is sliced into 2 slices. If you want know which pie has been eaten more, you make both pies have the same amount of slices, by slicing them into 6 pieces each.
Addition of mixed numbers
- Think of mixed numbers as a whole number AND a fraction. You can add whole numbers and you can add fractions with like denominators. Therefore, you can add mixed numbers!
- Add mixed numbers together, simplify
- First, add whole number part, then add fraction part
- 17 2/9 + 5 1/9
- Step 1: Expand out and re-write
- 17 + 2/9 + 5 + 1/9
- Step 2: Add whole numbers, then fractions (remember, the commumtative law of addition states order does not matter)
- (17 + 5) + (2/9 + 1/9)
- 22 + 3/9
- Step 3: simplify
- 22 1/3
Adding mixed numbers with unlike denominators
- Add mixed numbers together, simplify
- Add whole number part and then add fraction part
- If fractions are unlike, turn them into like fractions by finding LCM/common denom
- 3 1/12 + 11 2/5 + 4 3/15
- Step 1: Expand out, breaking mixed number into whole number and fraction part
- 3 + 1/12 + 11 + 2/5 + 4 + 3/15
- Step 2: Add whole numbers, then fractions (remember, the commumtative law of addition states order does not matter)
- (3 + 11 + 4) + (1/12 + 2/5 + 3/15)
- 18 + (1/12 + 2/5 + 3/15)
- Step 3: Can’t add fractions because they are unlike. Convert fractions to like by finding common denominator/LCM
- 18 + (5/60 + 24/60 + 12/60)
- 18 41/60
Adding mixed number with an improper fraction
- improper fraction = numerator larger then denominator
- there are 2 methods, method one: converting to improper fractions or method two: adding whole number part, then fraction part. We will use latter method
- Add whole number part and then fraction part, then convert improper fraction to proper
- 1 4/8 + 2 5/8
- step 1: Expand out, breaking mixed number into whole number and fraction part (rewrite)
- 1 + 4/8 + 2 + 5/8
- step 2: add whole numbers and fractions
- 3 + 9/8
- step 3: re-write improper fraction
- 3 + 1 1/8
- 4 + 1/8
- 4 1/8
Substracting mixed numbers
there are 2 methods, method one: converting to improper fractions or method two: adding whole number part, then fraction part. We will use latter method
- 5 5/8 - 2 1/5
- Step one: subtract whole numbers part and subtract fractions part
- Step two: convert fractions into like fractions by finding common denominator (a number which divides by both 8 and 5, the lCM)
3 17/40
note: if you are substracting a larger fraction from a smaller one, which will result in a negative fraction, it is better to do method one, where you go straight to an improper fraction for both of them
Substracting mixed numbers with negatives
When you are substracting a larger fraction from a smaller one, which will result in a negative fraction, it is better to do method one, where you go straight to an improper fraction for both of them
there are 2 methods, method one: converting to improper fractions or method two: adding whole number part, then fraction part. We will show both
8 2/3 - 5 5/6
- method one: converting to improper fractions straight away
- step one: convert to improper fraction
- 26/3 - 35/6
- step two: convert to like fraction (find common denominator, i.e. LCM)
- 52/6 - 35/6
- step three: substract and convert into mixed
- 52/6 - 35/6 = 17/6 = 2 5/6
- note: improper fraction method will also work but sometimes it can be a little harder or easier, depending on fractions
- method two: adding whole number part, then fraction part
- Step one: subtract whole numbers part and subtract fractions part
- 8 2/3 - 5 5/6
- (8-5) + (2/3 - 5/6)
- 3 + (2/3 - 5/6)
- Step two: convert fractions into like fractions by finding common denominator (the smallest number which divides by 3 and 6, the lCM)
- 3 + (4/6 - 5/6)
- we can’t subtract 4/6 - 5/6 because 5/6 is larger
- step three: re-group fraction to make 4/6 larger than 5/6*
- 2 + 1 + (4/6 - 5/6)
- 2 + 6/6 + (4/6 - 5/6)
- 2 + (10/6 - 5/6)
- 2 + 5/6
- method three: similar to method two but grouping takes place before adding whole numbers. Potentially easier
- Step one: re-write mixed numbers with their fractional parts and whole number parts and re-write the fractions with a common denominator
- 8 2/3 - 5 5/6
- (8-5) + (2/3 - 5/6)
- (8-5) + (4/6 - 5/6)
- we can’t subtract 4/6 - 5/6 because 5/6 is larger
- step two: re-group fraction to make 4/6 larger than 5/6*
- 8 4/6 can be regrouped so that the fractional part is greater than 5/6
- 8 4/6
- 8 + 4/6
- 7 + 1 + 4/6
- 7 + 6/6 + 4/6
- 7 + 10/6
- (7-5) + (10/6 - 5/6)
- 2 + 5/6
Chapter 8: Decimals
Decimals are useful in the real world for weighing and measuring things. Why? Because not everything is an exact whole number. For example, you might have 1 and 1/2 packets of polo, or 1.5 packets of polos.
- Remember, decimals show numbers less than one.
- For example, let’s say you measure a grain of sand with a ruler. It goes past 1 but doesn’t quite reach 2.
- Do we just give up? or take the number which is closest? No! Let’s split up the distance between 1 and 2 into tenths.
- Ok, so we can see it’s between 1.2 and 1.3, but it isn’t hitting eithr of them exactly.
- So, now do we give up? Nooooo!! Let’s split again, this time splitting the space between 1.2 and 1.3 into hundreths!
- Hey, it’s hit a point! The grain of sand is 1.25 long! Wo-hooooo.
- The key thing to remember is fractions is about splitting stuff up into smaller pieces to make it more precise when we measure. What will give you more precision when you sew? A thick clunky needle made of bone or a thin steel needle with a tiny sharp point? We get more precision and accuracy with smaller things and some things are too small to be measured by big numbers, just like bugs which are too small to be seen without a microscope!
Introduction to decimals
decimals are like fractions, they are a way of showing numbers less than one.
Everything beyond the decimal point is a fraction, a number less than one, and everything to the left of the decimal is a whole number

Tenths and Hundreds
writing decimals shown in grids

Comparing decimal place value
- Multiplying by a fraction moves each digit to the right
- multiplying by 1/10 moves each digit one value place to the right
- multiplying by 1/100 moves each digit 2 value places to the right
- Multiplying by a positive whole number moves each digit number to the left
- multipling by 10 moves each digit one value place to the left
- example: 83 hundreths x 1/10 = 83 thousandths
- multiplying by 1/10 moves each digit one value place to the right
- if we have 83 hundreths and we multiply them by 1/10, then each of the hundreths become a thousandths
- now instead of 83 hundreths we have 83 thousandths

Decimals on the number lines
- tens
- First, check the distance between 2 whole numbers and state by how many equal pieces it is divided by
- Next, state what each tick represents, i.e. 0.1 or 1 tenth
- In the example below the answer represents 6 ones and 2 tenths, or 62 tenths
- 1.9 is one-tenth less than 2.0. Understand now?

- hundreds
- The distance between two numbers is 1 tenth or 0.1
- This tenth is divided into 10 equal pieces. 0.1/10 = 0.01
- Each tick marks 0.01 or a hundreth

- negative decimals on a number line
- Remember, decimals show numbers less than one.
- Now would be a brilliant time to re-read the introduction to decimals above, it will help you understand the following graph
- To place a hundreths point, look at the graph below, remember it’s all about splitting up big numbers into smaller ones. Just keep going into we get to the hundreths place!

- To place a thousandths point, like -0.095, on a number do the following:
- step one: Look at each value place and place it.
- -0.095 is between 0 and -1
- Ones place: There is zero ones. So we’ll place it at zero as we can’t reach -1.
- tenths place: there is no tenths, so we can’t reach -0.1
- hundreds place: there is 9 hundreths. Let’s move it 9 ticks down the line. -0.095 is between -0.09 and -0.10
- thousandths place: there is 5 thousandths, that will be between the -0.09 and -0.10

Re-writing a fraction as a decimal
- Converting fractions into terms of tenths
- **IF DENOMINATOR IS A DIVISOR OF TEN: Multiply the denominator and numerator, i.e. turn the fraction into a term of tenths
- If the denominator is a divisor of ten, then multiply it to change it to ten
- example: 3/5 = 6/10 = 0 ones and 6 tenths or 0.6
- **IF DENOMINATOR IS NOT A DIVSOR OF TEN: Divide the denominator and numerator, then multiply into terms of tenths. In other words, Simplify then turn into terms of tenths. .
- If the denominator is not a divisor of ten, then simplyify the fraction by dividing the numerator and denominator, and then multiply then terms of tenths
- example: 6/12. GCF of 6 and 12 is 6. Simplify then turn into terms of tenths.
- 6/12 = 1/2 = 5/10 = 0 ones and 5 tenths or 0.5
- Another way of thinking is that 6 is half of 12, so what is half of 10? 5 is half of ten.
- example: 21/60. The GCF of 21 and 60 is three. Simplify then turn into terms of tenths.
- 21/60 = 7/20 = 35/100 = 0.35 or 3 tenths and 5 hundreths
- Divide the numerator by denominator, then multiply in terms of tenths
- The quickest and easiest way
- example: 6/12 = 6 divided by 12 = 0.5 or zero ones and 5 tenths
Decimal to simplified fraction
write as a mixed number (whole number parts and fraction parts), convert the whole number parts into fractions, add fractions together, simplify
- example: Write 2.75 as a simplified fraction.
- Write as mixed number: 2 + 7/10 + 5/100
- Add Fraction parts: 2 + 70/100 + 5/100 = 2 75/100
- to add fractions you need a common denominator. 100 is a common denomiator of both fractions
- **2 options: (A) Keep as simplified mixed number, (B) turn into a simplified fraction
- A: 2 75/100 = 2 3/4
- B: 2 3/4 = 1 + 1 + 3/4 = 4/4 + 4/4 + 3/4 = 11/4
- turn whole numbers into fraction (2 = 1 + 1 = 4/4 + 4/4), add together as they have the same denom
- 2.75 is the same thing as 2 and 75-hundreths which is the same thing as 2 and 3-fourths
- 2.75 = 2 75/100 = (A) 2 3/4 = (B) 11/4
- stopping at answer A, as a mixed number, is acceptable
Writing a fraction as a decimal
- step one: re-write as fraction using place value
- 0.8 = 8/10. There is an 8 in the tenTHS place, so we have 8-tenths
- step two: simplify
- 8/10 = 4/5. 2 is a common factor which goes into both 8 and 10.
- 0.8 = 8/10 = 4/5. zero point eight is the same thing as eight-tenths which is the same thing as fourth-fifths
- example: write 0.36 as a fraction
- 0.36 = 36/100.
- There is a 3 in the tenths place, so we have 3-tenths, 3-tenths can be written as 3/10
- There is a 6 in the hundreths place, so we have 6-hundreths, 6-hundreths can be written as 6/100
- The sum of these two is 30/100 + 6/100 = 36/100
- 0.36 = 36/100 = 9/25. Numerator and Denom share a common factor, so we reduce…GCF(36,100) = 4
Comparing Decimals
- To compare decimals you can:
- Compare the largest place value
- What is the highest place value where the numbers have different digits?
- 0.2 & 0.17. Largest place value is tenths, 0.2 has more tenths (2-tenths) than 0.17 (1-tenth) which means 0.2 > 0.17
- If the values were equal in the tenths place you would keep moving down the place value list
- Example Compare 0.7 and 0.09
- 0.7 > 0.09
- The place value, column, is more important than the number
- Although 9 is bigger than 7, it’s in a smaller place value!
- Even if you increased 9-hundreths (0.09) by another 1-hundreth (0.01), you’d only get 1-tenth (0.1) which is still smaller than 7-tenths (0.7)
- Another way. Turn into like fractions: Compare 0.7 and 0.09
- 70-hundreths (70/100, 0.70) is larger than 9-hundreths (9/100, 0.9)
- Compare 0.31 and 0.29. 31-hundreths (31/100) is larger than 29-hundreths (29/100)
- Summary:
- Compare 0.8 and 0.09
- 0.8 > 0.09 because…
- Method one - largest place value: 8-tenths (8/10) is larger than zero-tenths (0/10). 0.09 has fewer tenths than 0.8.
- Method two - like fractions: 80-hundreths (80/100) is larger than 9-hundreths (9/100) 0.09 has fewer hundreths than 0.80
- Compare 9.75 and 9.49
- 9.75 > 9.49 cause…
- Method 3 - mixed number: 9 + 7/10 > 9 + 4/10
- 9 pies and 7 slices is more than 9 pies and 4 slices

- useful tip: When comparing numbers with different places values it is useful to add in zero
- Compare 400.1 and 58.4
- re-write as 400.1 and 058.4
- Compare 0.23 and 0.4

- note: Remember that the larger number shoots the smaller one, 0.8 > 0.22
Comparing Decimals visually
- The decimal which covers most area is largest

Compare on number line
- Smallest decimals are to the left on the number line
Using Decimals
Converting Units: centimeters to meters
Centi means 100th, so centimeter = 1/100 metre
- To turn centimeter into a meter you divide by 100
- reality check: After converting a centimeter into a meter, should it be a larger or smaller number? Smaller! Meters is a larger unit. After converting cm to m, you will always have a fewer number of meters than centimeters, so the number should be smaller.

Subtraction of decimals

Mensuration
Area
Area = l * w
- square metre = the amount of squares you can fit into a shape
- a field is 12m squared, which means you can fit 12 squares (that are 1mx1m) into it
- If a rectangle has a width of 4 units and length of 6 units, then a column of 4 squares will fit into the rectangle 6 times.

Area and distributive property
- To find the area of a shape you can split the shape into smaller shapes, find the area of those shapes and sum them together
- i.e. split a rectangle into two smaller rectangles.

Find the length of a shape using area
- square
- find the square root of the area
- A square has an area of 100 units sq. What is the length of one side?
- 25 * 25 = 100
- Each side is 25 units
Chapter 11: Algebra
Origins
- Algebra is a way of working with unknown numbers or numbers that don’t exist
This is a great leap, because in the same way that numbers allowed you to count things that aren’t physically there (useful when you need to figure out how much food to hunt or sheep you lost), algebra let’s your mind do maths with your imagination. For example, a 10% on a $10 shirt is $1. I wonder how much of a discount I would get if it was a $20 shirt? In comes algebra
- history
- 2000bc - bablyonians
- 200ce - diophantus
- 600ce - Brahmagupta
- 800ce - Al-kwarazi, whose book is where we get the word algebra
The Idea of a variable, expression and equation
- variable = a number that varies between scenarios
- 10 + tips
- 10 + 4 = 14. First day of work
- 10 + 8 = 18. Working on Christmas day
- expression = any number or letter/variable that are together in a operation
- equation = two expressions that are equal to each other
- 5 + 3 = 6 + 2
- 5 + 3 is an expression and 6 + 2 is an expression. Both expressions equal 8, so are equal
- x + 3 = 1.
- x + 3 is an expression and 1 is an expression
- x + 2 + 3 = 5.
- x + 2 + 3 is one expression and 5 is another. Both expressions are equal
- True Equation = equation with only numbers in it. 5 + 3 = 6 + 2
- Algebraic Equation = equation with a variable. x + 3 = 1
- Finding a solution = turning a algebraic equation into a true equation
- Finding out what x is, for example:
- What is x?
- x + 2 + 2 = 5
- x + 5 = 5
- x = 0. The solution to x is zero!
Expressions with variables: Evaluating and writing expressions!
Evaluating an expression with one variable
To value an expression with a variable we use a technique called subsitution (or “pluggin in”), where you replace the variable name with a value
The first thing to do when evaluating expressions is to give the variable a value
- Ash has entered the Pokemon safari and will win a cash prize depending on which Pokemon he catches. Evaluate the following expression, a + 5, where variable a = the pokemon’s level.
- To evaulate the following expression we need to plug in a value for the variable
a. Evaluate a + 5 when a =…
- a = 2. 2 + 5 = $7
- a = 53. 53 + 5 = $58
- a = 90. 90 + 5 = $95
Expression value intuition
- What happens to your expression as the value of the variable changes?
- 100 - x. What happens to the expression if x is increasing?
- As x decreases, the expression decreases
- 5 / x + 5. What happens to the expression if x is decreasing?
- As x decreases, the expression increases
- 3y / 2y. What happens to the expression if y is increasing and is positive?
- Stays the same. y and y are the same value, so y over y cancel out to 1
Writing expressions
Writing an expression is changing words like, “three more than x” to x + 3
But why do we change words into mathematical expressions? One of the reasons is maths is more precise and easier to work with. Once you’ve written something down as a math’s expression you can have a lot of fun
- Use
- Algebraic expressions are used widely in every-day life, from converting units, calculating costs and writing computer programs
- For example, every time you press a button Mario might move two steps.
b + 2 = m. b = amount of times button is pressed, m = total amont of steps mario takes
- example: Take the quantatity -3 times x, and then plus 1
Writing basic expression word problems
- Ged brought x staffs for 120 gold. How much gold did he spend?
- Pikachu’s thunderbolt does 2 damage for each of his levels. If it is a critical hit then 8 is added onto the total damage. What’s the total damage Pikachu does if its a critical hit?
Equations and inequalities
Testing solutions to equations
- remember that an equation = a statement that two expressions are equal
- The equation above only includes numbers, but if an equation includes a variable it is a algebraic equation
- We find out how a variable is by solving it. In other words, solving is turning a algebraic equation into a true equation
One-step equation intuition
Dividing Both sides of an Equation
One-step equation addition and subtraction
*Now we’re comfortable with the why of why we do something to both sides (to keep it equal). Let’s apply it to solve for a unknown variable (x)
We know that we do the same thing to both sides of an equation, but how?
One method of solving an equation (turning an algebraic equation into a true one) is to add or subtract both sides
- To solve an equation, all you want on the left handside is the variable (x)
- x + 7 = 12.
- The above equation says we have a variable x which plus 7 is equal to 12. Remove 7 to be left with variable (x) on left handside
- x - 7 = 12 - 7. Doth same things to both sides
- x = 5
Summary of how to solve addition and subtraction equations
To solve for x you need to undo the addition or subtraction. How? Do the inverse operation
| x + 20 = 45 |
addition equation |
subtract 20 from each side |
| p - 23 = 24 |
subtraction equation |
add 23 to each side |
one-step addition and subtraction: Fractions and Decimals
- it’s the same as for whole numbers. Solve an equation by getting the variable on its own. How? Do the same things to the both sides. Like? Add or subtract!
| x + 1/5 = 6/5 |
addition equation |
subtract 1/5 from each side |
| x - 18 = 30.4 |
subtraction equation |
add 18 to each side |
One step equation multiplication and divison
- To solve for
x you need to undo the multiplying or division
- how do you undo? Do the inverse! The inverse of addition is subtraction, the inverse of multiplication is division
| 3x = 21 |
multiplication equation |
divide each side by 3 / group each side into threes |
| x/2 = 24 |
division equation |
multiply each side by 2 |
